The following fact has been proved under various conditions: If a differential operator P generates a positivity preserving \(C_ 0\)- semigroup (on a certain function space), then P is of order at most 2 and (possibly degenerate) elliptic. See, e.g., K. Yosida, ”Functional analysis” (1980; Zbl 0435.46002), Chap. XIII, or M. Fukushima, ”Dirichlet forms and Markov processes” (1980; Zbl 0422.31007). The latter deals with the case of \(L^ 2\) spaces and contraction semigroups, although the generator considered is more general.
One of the purposes of this paper is to give an elementary and transparent proof of the above fact for the case of the space \(L^ p(R^ n)\) and (not necessarily contractive) positivity preserving \(C^ 0\)-semigroups (Theorem 3.6). The proof depends on the abstract Kato’s inequality for the generator of a positivity preserving \(C_ 0\)- semigroup on a Banach lattice and a change of variables argument. This result is generalized by the first named author in a succeeding paper contained in Aspects of positivity in functional analysis, North-Holland Math. Stud. 122, 241-246 (1986; Zbl 0608.47043).
The second purpose of this paper is to show that a certain second order elliptic differential operator with singular 0-th order therm generates a positivity preserving \(C_ 0\)-semigroup on \(L^ p(R^ n)\). The result depends on a perturbation theorem obtained by the second named author.